The solution shown in figure REF  is marched downstream using the PFE method, which allows us to obtain the modulated wavepackets supported by the shock-containing mean flow at each frequency. Figures REF  and REF  show, respectively, streamwise velocity and pressure fields obtained using PFE for \(St=0.4\)  and \(0.7\) , where both the real part (top) and absolute value (bottom) are shown to highlight the oscillatory behaviour and the modulation of the wavepackets. In these figures, the first row (a,b) depicts the total field obtained from PFE, and the subsequent ones show components \(\mathbf {\hat{q}_{0}}\)  (c,d), \(\mathbf {\hat{q}_{+1}}\)  (e,f) and \(\mathbf {\hat{q}_{-1}}\)  (g,h). Overall, the structures are very similar to previous PSE results (see [1]}, for instance): the initial KH solution grows exponentially in the early stations of the jet, then it saturates and decays as the shear-layer thickens further downstream. While the oscillatory behaviour and the growth/decay of the solution is similar to the most energetic structures in subsonic and ideally expanded supersonic jets [2]}, some features of the solutions are unique to PFE, and these are better seen in the absolute value of the reconstructed fields in figures REF  and REF . In all cases, the amplitude envelope of the wavepacket has an oscillatory behaviour, especially close to the centreline (where the shock-cell structure has maximum amplitude); these oscillations are associated with the modulation wavenumbers included in the spatial march which, for this case, will lead to the high-wavenumber structures observed in the total field (due to the number of harmonics considered in the analysis).
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